R E S E A R C H
Open Access
Strong convergence for asymptotically
nonexpansive mappings in the intermediate
sense
Gang Eun Kim
**Correspondence:
kimge@pknu.ac.kr Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Korea
Abstract
In this paper, letCbe a nonempty closed convex subset of a strictly convex Banach space. Then we prove strong convergence of the modified Ishikawa iteration process whenTis an ANI self-mapping such thatT(C) is contained in a compact subset ofC, which generalizes the result due to Takahashi and Kim (Math. Jpn. 48:1-9, 1998).
MSC: 47H05; 47H10
Keywords: strong convergence; fixed point; Mann and Ishikawa iteration process; ANI
1 Introduction
LetCbe a nonempty closed convex subset of a Banach spaceE, and letTbe a mapping of
Cinto itself. ThenTis said to beasymptotically nonexpansive[] if there exists a sequence
{kn},kn≥, withlimn→∞kn= , such that
Tnx–Tny≤knx–y
for allx,y∈Candn≥. In particular, ifkn= for alln≥,Tis said to benonexpansive. Tis said to beuniformly L-Lipschitzianif there exists a constantL> such that
Tnx–Tny≤Lx–y
for allx,y∈Candn≥.Tis said to beasymptotically nonexpansive in the intermediate sense(in brief, ANI) [] providedTis uniformly continuous and
lim sup
n→∞
sup
x,y∈C
Tnx–Tny–x–y≤.
We denote byF(T) the set of all fixed points ofT,i.e.,F(T) ={x∈C:Tx=x}. We define the modulus of convexity for a convex subset of a Banach space; see also []. LetCbe a nonempty bounded convex subset of a Banach spaceEwithd(C) > , whered(C) is the diameter ofC. Then we defineδ(C,) with ≤≤ as follows:
δ(C,) =
rinf
maxx–z,y–z–z–x+y
:x,y,z∈C,x–y ≥r
,
wherer=d(C). When{xn}is a sequence inE, thenxn→xwill denote strong convergence
of the sequence{xn}tox. For a mappingsT ofCinto itself, Rhoades [] considered the
following modified Ishikawa iteration process (cf.Ishikawa []) inCdefined byx∈C:
xn+=αnTnyn+ ( –αn)xn, yn=βnTnxn+ ( –βn)xn, (.)
where{αn}and{βn}are two real sequences in [, ]. Ifβn= for alln≥, then the iteration
process (.) reduces to the modified Mann iteration process [] (cf.Mann []).
Takahashi and Kim [] proved the following result: LetE be a strictly convex Banach space andCbe a nonempty closed convex subset ofEandT:C→Cbe a nonexpansive mapping such thatT(C) is contained in a compact subset ofC. Supposex∈C, and the sequence{xn}is defined byxn+=αnT[βnTxn+ ( –βn)xn] + ( –αn)xn, where{αn}and
{βn}are chosen so thatαn∈[a,b] andβn∈[,b] orαn∈[a, ] andβn∈[a,b] for somea,b
with <a≤b< . Then{xn}converges strongly to a fixed point ofT. In , Tsukiyama
and Takahashi [] generalized the result due to Takahashi and Kim [] to a nonexpansive mapping under much less restrictions on the iterative parameters{αn}and{βn}.
In this paper, letCbe a nonempty closed convex subset of a strictly convex Banach space. We prove that ifT:C→Cis an ANI mapping such thatT(C) is contained in a compact subset ofC, then the iteration{xn}defined by (.) converges strongly to a fixed point ofT,
which generalizes the result due to Takahashi and Kim [].
2 Strong convergence theorem We first begin with the following lemma.
Lemma .[] Let C be a nonempty compact convex subset of a Banach space E with r=d(C) > .Let x,y,z∈C and supposex–y ≥r for some with≤≤.Then,for allλwith≤λ≤,
λ(x–z) + ( –λ)(y–z)≤maxx–z,y–z– λ( –λ)rδ(C,).
Lemma .[] Let C be a nonempty compact convex subset of a strictly convex Banach space E with r=d(C) > .Iflimn→∞δ(C,n) = ,thenlimn→∞n= .
Lemma .[] Let{an}and{bn}be two sequences of nonnegative real numbers such that
∞
n=bn<∞and an+≤an+bn
for all n≥.Thenlimn→∞anexists.
Lemma . Let C be a nonempty compact convex subset of a Banach space E,and let T:C→C be an ANI mapping.Put
cn=sup x,y∈C
Tnx–Tny–x–y∨,
Proof The existence of a fixed point ofT follows from Schauder’s fixed theorem []. For a fixedz∈F(T), since
Tnyn–z≤ yn–z+cn
=βnTnxn+ ( –βn)xn–z+cn
≤βnTnxn–z+ ( –βn)xn–z+cn
≤βnxn–z+cn+ ( –βn)xn–z+cn
≤ xn–z+ cn,
we obtain
xn+–z=αnTnyn+ ( –αn)xn–z
≤αnTnyn–z+ ( –αn)xn–z
≤αn
xn–z+ cn
+ ( –αn)xn–z
≤ xn–z+ cn.
By Lemma ., we readily see thatlimn→∞xn–zexists.
Theorem . Let C be a nonempty compact convex subset of a strictly convex Banach space E with r=d(C) > .Let T:C→C be an ANI mapping.Put
cn=sup x,y∈C
Tnx–Tny–x–y∨,
so that∞n=cn<∞.Suppose x∈C,and the sequence{xn}defined by(.)satisfiesαn∈
[a,b]andlim supn→∞βn=b< orlim infn→∞αn> andβn∈[a,b]for some a,b with
<a≤b< .Thenlimn→∞xn–Txn= .
Proof The existence of a fixed point ofT follows from Schauder’s fixed theorem []. For any fixedz∈F(T), we first show that ifαn∈[a,b] andlim supn→∞βn=b< for some a,b∈(, ), then we obtain limn→∞Txn–xn= . In fact, letn=T
nyn–x n
r . Then we
have ≤n≤ sinceTnyn–xn ≤r. As in the proof of Lemma ., we obtain
Tnyn–z≤ xn–z+ cn. (.)
Since
Tnyn–xn=rn,
and by (.) and Lemma ., we have
xn+–z=αn
Tny n–z
+ ( –αn)(xn–z)
Thus
αn( –αn)rδ(C,n)≤ xn–z–xn+–z+ cn.
Since
r ∞
n=
a( –b)δ C,T
ny n–xn
r
<∞,
we obtain
lim
n→∞δ C,
Tny n–xn
r
= .
By using Lemma ., we obtain
lim
n→∞T ny
n–xn= . (.)
Since
Tnxn–xn≤Tnxn–Tnyn+Tnyn–xn
≤ xn–yn+cn+Tnyn–xn
=βnTnxn–xn+cn+Tnyn–xn,
we obtain
( –βn)Tnxn–xn≤cn+Tnyn–xn. (.)
Sincelim supn→∞βn=b< , we have
lim inf
n→∞( –βn) = –b> . (.)
From (.), (.) and (.), we obtain
lim
n→∞T nx
n–xn= . (.)
Since
xn+–xn=( –αn)xn+αnTnyn–xn
=αnTnyn–xn
≤bTnyn–xn,
and by (.), we obtain
lim
Since
xn–Txn
≤ xn–xn++xn+–Tn+xn++Tn+xn+–Tn+xn+Tn+xn–Txn
≤xn–xn++cn++xn+–Tn+xn++Tn+xn–Txn
and by the uniform continuity ofT, (.) and (.), we have
lim
n→∞xn–Txn= . (.)
Next, we show that iflim infn→∞αn> andβn∈[a,b], then we also obtain (.). In fact, let n=T
nxn–x n
r . Then we have ≤n≤. Fromlim infn→∞αn> , there are some positive
integernand a positive numberasuch thatαn>a> for alln≥n. Since
xn+–z=αn
Tnyn–z
+ ( –αn)(xn–z)
≤αnTnyn–z+ ( –αn)xn–z
≤αnyn–z+αncn+ ( –αn)xn–z,
and hence
xn+–z–xn–z αn
≤ yn–z–xn–z+cn.
So, we obtain
xn–z–yn–z ≤
xn–z–xn+–z
αn
+cn
≤xn–z–xn+–z
a +cn. (.)
Since
Tnxn–z≤ xn–z+cn,
from Lemma ., we obtain
yn–z=βnTnxn+ ( –βn)xn–z
=βn
Tnxn–z
+ ( –βn)(xn–z)
≤ xn–z+cn– βn( –βn)rδ(C,n). (.)
By using (.) and (.), we obtain
βn( –βn)rδ(C,n)≤ xn–z–yn–z+cn
≤xn–z–xn+–z
Hence
r ∞
n=
a( –b)δ C,T
nx n–xn
r
<∞.
We also obtain
lim
n→∞xn–T nx
n= (.)
similarly to the argument above. Since
yn–xn=βnTnxn+ ( –βn)xn–xn
≤βnTnxn–xn
≤bTnxn–xn,
and by using (.), we obtain
lim
n→∞xn–yn= . (.)
Since
Tnyn–xn≤Tnyn–Tnxn+Tnxn–xn
≤ yn–xn+cn+Tnxn–xn,
by using (.) and (.), we obtain
lim
n→∞T ny
n–xn= . (.)
Since
Tnyn–yn≤Tnyn–xn+xn–yn,
by using (.) and (.), we obtain
lim
n→∞T ny
n–yn= . (.)
Since
xn–xn–=( –αn–)xn–+αn–Tn–yn––xn– =αn–Tn–yn––xn–
≤Tn–yn––yn–+yn––xn–,
by (.) and (.), we get
lim
From
Tn–xn–xn
≤Tn–xn–Tn–xn–+Tn–xn––xn–+xn––xn
≤xn–xn–+cn–+Tn–xn––xn–
and by (.) and (.), we obtain
lim
n→∞T n–x
n–xn= . (.)
Since
xn–Txn
≤ xn–yn+yn–Tnyn+Tnyn–Tnxn+Tnxn–Txn
≤yn–Tnyn+ xn–yn+cn+Tnxn–Txn
and by the uniform continuity ofT, (.), (.) and (.), we have
lim
n→∞xn–Txn= .
Our Theorem . carries over Theorem of Takahashi and Kim [] to an ANI mapping.
Theorem . Let C be a nonempty closed convex subset of a strictly convex Banach space E,and let T:C→C be an ANI mapping,and let T(C)be contained in a compact subset of C.Put
cn=sup x,y∈C
Tnx–Tny–x–y∨,
so that∞n=cn<∞.Suppose x∈C,and the sequence{xn}defined by(.)satisfiesαn∈
[a,b]andlim supn→∞βn=b< orlim infn→∞αn> andβn∈[a,b]for some a,b with
<a≤b< .Then{xn}converges strongly to a fixed point of T.
Proof By Mazur’s theorem [],A:=co({x} ∪T(C)) is a compact subset ofC contain-ing{xn}which is invariant underT. So, without loss of generality, we may assume that C is compact and{xn}is well defined. The existence of a fixed point ofT follows from
Schauder’s fixed theorem []. Ifd(C) = , then the conclusion is obvious. So, we assume
d(C) > . From Theorem ., we obtain
lim
n→∞xn–Txn= . (.)
SinceCis compact, there exist a subsequence{xnk}of the sequence{xn}and a pointp∈C such thatxnk →p. Thus we obtainp∈F(T) by the continuity ofT and (.). Hence we
Corollary . Let C be a nonempty closed convex subset of a strictly convex Banach space E,and let T:C→C be an asymptotically nonexpansive mapping with{kn} satisfy-ing kn≥,
∞
n=(kn– ) <∞,and let T(C)be contained in a compact subset of C.Suppose x∈C,and the sequence{xn}defined by(.)satisfiesαn∈[a,b]andlim supn→∞βn=b< orlim infn→∞αn> andβn∈[a,b]for some a,b with <a≤b< .Then{xn}converges strongly to a fixed point of T.
Proof Note that
∞
n=
cn= ∞
n=
(kn– )diam(C) <∞,
where diam(C) =supx,y∈Cx–y<∞. The conclusion now follows easily from
Theo-rem ..
We give an example which satisfies all assumptions ofTin Theorem .,i.e.,T:C→C
is an ANI mapping which is not Lipschitzian and hence not asymptotically nonexpansive.
Example . LetE:=RandC:= [, ]. DefineT:C→Cby
Tx=
, x∈[, ];
√
–x, x∈[, ].
Note thatTnx= for allx∈Candn≥ andF(T) ={}. Clearly,Tis uniformly continuous,
ANI onC, butTis not Lipschitzian. Indeed, suppose not,i.e., there existsL> such that
|Tx–Ty| ≤L|x–y|
for allx,y∈C. If we takey:= andx:= –(L+) > , then
√
–x≤L( –x) ⇔
L ≤ –x=
(L+ ) ⇔ L+ ≤L. This is a contradiction.
We also give an example of an ANI mapping which is not a Lipschitz function.
Example . LetE=RandC= [–π, π] and let|h|< . LetT:C→Cbe defined by
Tx=hxsinnx
for eachx∈Cand for alln∈N, whereNdenotes the set of all positive integers. Clearly
F(T) ={}. Since
T(x) =hxsinnx,
we obtain{Tnx} → uniformly onCasn→ ∞. Thus
lim sup
n→∞
Tnx–Tny–x–y ∨=
for allx,y∈C. HenceT is an ANI mapping, but it is not a Lipschitz function. In fact, suppose that there existsh> such that|Tx–Ty| ≤h|x–y|for allx,y∈C. If we take
x=π
n andy=
π
n, then
|Tx–Ty|=hπ
nsinn
π
n–h
π
nsinn
π
n
=hπ
n ,
whereas
h|x–y|=hπ
n –
π
n
=hπ
n .
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author would like to express their sincere appreciation to the anonymous referee for useful suggestions which improved the contents of this manuscript.
Received: 13 February 2014 Accepted: 4 July 2014 Published: 23 July 2014
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doi:10.1186/1687-1812-2014-162
Cite this article as:Kim:Strong convergence for asymptotically nonexpansive mappings in the intermediate sense.
